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Angiogenesis is the process by which new blood vessels develop from an existing vasculature, through endothelial cell sprouting, proliferation and fusion. Adult endothelial cells are normally quiescent and, apart from certain developmental processes (e.g. embryogenesis) and wound healing, angiogenesis is generally a pathological process implicated in arthritis, certain eye diseases and solid tumour development, invasion and metastasis. Tumour-induced angiogenesis occurs when a small avascular tumour exceeds some critical diameter (~2 mm), above which normal tissue vasculature is no longer able to support its growth. At this stage, the tumour cells lacking nutrients and oxygen become hypoxic. This is assumed to trigger cellular release of tumour angiogenic factors (TAF), which start to diffuse into the surrounding tissue and approach the endothelial cells (EC) of nearby blood vessels. Endothelial cells subsequently respond to the TAF concentration gradient by forming sprouts, migrating and proliferating towards the tumour. It takes approximately 10 to 21 days for the growing network to link the tumour to the parent vessel, and this vascular connection subsequently provides all the nutrients and oxygen required for continued tumour growth [Paweletz and Knierim, 1989]

Interest in the mathematical modelling of blood vessel growth and development may be traced back nearly a century to Dundee's very own Sir D'Arcy Wentworth Thompson who, in his book "On Growth and Form", devotes a section entitled "On the Form and Branching of Blood Vessels" and considers "...a number of interesting points in connection with the form and structure of the blood-vessels."

As D'Arcy Thompson also notes here:

"Many problems of a hydrodynamical kind arise in connection with the flow of blood through the blood-vessels; and while these are of primary importance to the physiologist they interest the morphologist in so far as they bear on the questions of structure and form. As an example of such mechanical problems we may take the conditions which go to determine the manner of branching of an artery, or the angle at which its branches are given off;…This is a vastly important theme… and helps to bring the morphological and the physiological concepts together."

Modelling work over the years by members of the Dundee Mathematical Biology group, along with collaborators, has focussed on the aspect of endothelial cell migration and capillary formation as well as on blood and drug flow through the capillary network. The group has developed novel mathematical modelling techniques (hybrid discrete-continuum models) to model the growth and development of new capillaries [Anderson and Chaplain, 1998]. The work of group members goes back to the early 1990s.

By contrast, blood flow in a tumour-induced (micro) capillary network has only been modelled relatively recently. Blood is a complex fluid, the rheological properties of which lead to interesting feedback mechanisms during perfusion. For example, shear stresses generated within the capillary network by the flowing blood strongly influence vessel adaptation and network remodelling. These shear stresses are, in turn, affected by blood viscosity, the distribution of which depends upon a non-uniform distribution of haematocrit (volume fraction of red blood cells contained in the blood) within the host vasculature (the Fahraeus effect). However, the distribution of haematocrit itself depends upon the spatial architecture of the underlying network and so the feedback is established, "the modeling loop is closed", so to speak.

The group is currently developing a dynamic, adaptive model of tumour-induced angiogenesis.

Key group publications:

  • A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor,
    Chaplain, M.A.J., Stuart, A.M., IMA J. Math. Appl. Med. Biol., 10: 149-68 (1993)

  • Mathematical models for tumour angiogenesis: numerical simulations and nonlinear wave solutions,
    Byrne, H.M., Chaplain, M.A.J., Bull. Math. Biol., 57: 461-486 (1995)

  • Two-dimensional models of tumour angiogenesis and anti-angiogenesis strategies
    Orme, M.E., Chaplain, M.A.J., IMA J. Math. Appl. Med. Biol., 14: 189–205 (1997)

  • Continuous and discrete mathematical models of tumor-induced angiogenesis,
    Anderson, A.R.A., Chaplain, M.A.J., Bull. Math. Biol., 60: 857-99 (1998)

  • Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies,
    McDougall, S.R., Anderson, A.R.A., Chaplain, M.A.J., Sherratt, J.A., Bull. Math. Biol., 64: 673-702 (2002)

  • Mathematical modelling of flow in 2D and 3D vascular networks: applications to anti-angiogenic and chemotherapeutic drug strategies,
    Stephanou, A., McDougall, S.R., Anderson, A.R.A., Chaplain, M.A.J., Math. Comput. Modell., 41: 1137-1156 (2005)

  • Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies,
    McDougall, S., Anderson, A.R.A., Chaplain, M.A.J., J. Theor. Biol., 241: 564-589 (2006)

  • Mathematical modelling of the influence of blood rheological properties upon adaptive tumour-induced angiogenesis,
    Stephanou, A., McDougall, S., Anderson, A., Chaplain, M.A.J., Math. Comput. Modell., 44: 96-123 (2006)

  • Mathematical modelling of tumor-induced angiogenesis,
    Chaplain, M.A.J., McDougall, S.R., Anderson, A.R.A., Annu. Rev. Biomed. Eng. 8: 233-257 (2006)

  • Dynamics of angiogenesis during wound healing: a coupled in vivo and in silico study,
    Machado, M.J.C., Watson, M.G., Devlin, A.H., Chaplain, M.A.J., McDougall, S.R., Mitchell, C.A., Microcirculation, 18: 183-197 (2011)


  • Tumor-related angiogenesis,
    Paweletz, N., Knierim, M., Crit. Rev. Oncol. Hematol. 9: 197-242 (1989)

  • On Growth and Form,
    Thompson, D.W., Cambridge University Press, Cambridge (1917)